Theorem int0 4905

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
int0	⊢ ∩ ∅ = V

Proof of Theorem int0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4439	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥
2		vex 3434	. . . . 5 ⊢ 𝑦 ∈ V
3	2	elint2 4897	. . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥)
4	1, 3	mpbir 231	. . 3 ⊢ 𝑦 ∈ ∩ ∅
5	4, 2	2th 264	. 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V)
6	5	eqriv 2734	1 ⊢ ∩ ∅ = V

Syntax hints:   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430  ∅c0 4274  ∩ cint 4890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3432  df-dif 3893  df-nul 4275  df-int 4891

This theorem is referenced by:  unissint  4915  uniintsn  4928  rint0  4931  intex  5286  intnex  5287  oev2  8458  fiint  9237  elfi2  9327  fi0  9333  cardmin2  9923  00lsp  20976  cmpfi  23373  ptbasfi  23546  fbssint  23803  fclscmp  23995  zarcmplem  34025  rankeq1o  36353  bj-0int  37413  heibor1lem  38130  ipoglb0  49463  mreclat  49466